In Chapter 3 we saw that Basel II presented a generally lower capital charge for SMEs. However, this lower capital setting for SME borrowers, and in particular, the mechanisms through which it has been implemented, has come under scrutiny in the portfolio credit risk modelling and management literature. For example, Jacobson, Linde, and Roszbach (2005) and Dietsch and Petey (2004) test some of the SME assumptions made in Basel II on specially constructed data sets of aggregated credit portfolios. Despite the breadth of these data sets, however, the time series over which probabilities of default and correlations are measured have a generally restricted span.
Chapter 2 presented a general abundance of default data in the Financing Company’s historical portfolio. Here we construct additional segmentations to those observed in Chapter 2 and use them to explore the SME relationships modeled in Basel II. In particular, we use dual segmentations to estimate probabilities of default and correlations for various Size and Risk Rating segmentations.
This depth of SME default data is then used in a calibration exercise wherein explicit segmentations of borrowers according to size and credit quality allow for direct tests of Basel II pre-calibrations and assumed relationships. This work is similar both in spirit and technique to that of Dietsch and Petey (2004). We use the internal calibration methods of the Gordy (2000) single-factor portfolio credit risk model to measure asset
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correlations by risk and size segment, and explore the relationship between correlations – and by extension portfolio credit risk charges – along these borrower dimensions.
The use of this technique is doubly informative given the genesis of the Basel II IRB model as an implementation of work by Merton (1974), Vasicek (2002), and Gordy (2003). In addition, the estimation of asset correlations under various calibrations within a single-factor credit risk framework provides an easily implementable and comparable avenue along which to explore SME portfolio credit risk portfolio characteristics. To that end, our results in this Chapter demonstrate that for the Financing Company SME portfolio, the relationship between asset correlation and size, as well as those between asset correlation and probability of default, can differ to a large extent from those pre- calibrated in the Basel II portfolio credit risk mechanism.
In estimating asset correlations, greater emphasis is placed on the data quality of the time series over which defaults are counted. In particular, it becomes imperative that the default series be adequately populated, with both healthy and defaulted borrowers, such that an accurate understanding of default behaviour in a given segment is achievable. To that end we introduce several amalgamations of our previously defined Risk Ratings and Size Buckets, refered to as Risk Groups and Size Groups. The aim of these amalgamated groups is to bolster our estimation of correlations and their relationship with other credit characteristics among SME borrowers. In order to ensure an unbiasedness in our construction of Risk Groups we present several definitions and use cross comparisons between them as a check on the robustness of our conclusions.
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Having estimated asset correlations reflective of the SME credit characteristics found in the Financing Compnary portfolio, we apply them in the estimation of the portfolio credit risk as defined by the value-at-risk (VaR) and Economic Capital (EC) – or Credit-VaR (CVaR) – at a given confidence level.
In particular, EC values are generated in both asymptotic and non-asymptotic implementations of a single-factor framework; here asymptotic refers to a portfolio with a large number of borrowers such that idiosyncratic risks are assumed to be sufficiently diversified so as not to contribute to portfolio risk. This approximating asymptotic portfolio loss distribution is shown to hold even if borrower exposures are not uniform but with a large number of borrowers not one or a few of which are significantly larger than the rest Vasicek (2002).. The asymptotic framework is directly comparable to a stripped down – or naïve – version of the Basel IRB model which is itself based on an asymptotic single factor model; Gordy (2003) showed that the asymptotic single factor model employed in the regulatory capital mechanism is uniquely able to provide a portfolio invariant framework for capital calculation, a necessary condition for regulators looking to apply a consistent standardized model across varying financial institutions. The non-asymptotic implementation uses Monte Carlo simulations to generate and allocate capital charges while taking into account explicit idiosyncratic risks present in the portfolio. The juxtopostion of these asymptotic and non-asymptotic implementations within the same framework yields an interesting comparison and discussion of the “granularity effect” due to the application of an asymptotic model to a finite real-world
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SME portfolio. This granularity effect has been broadly discussed and dealt with in Gordy and Lutkebohmert (2007), BCBS (2006b), Tarashev and Zhu (2007), and other notable papers. Tarashev and Zhu (2007) perform an analysis similar to that performed in this paper.
Our analysis in this Chapter therefore focuses on two central aspects of portfolio credit risk modelling while using Basel II minimum capital regulations as a backdrop. The first is the estimation of correlations as they relate to credit events in an SME portfolio. The second focus of this Chapter is on capital charges generated by an SME portfolio using internally estimated asset correlations.
Our results here will show that, contrary to Basel prescriptions, asset correlations can not be shown to increase with probability of default, nor can they be shown to strictly increase with size within SME segments. These results compare starkly with findings of a generally increasing relationship between asset correlations and PDs in Dietsch and Petey (2004), Duellmann and Scheule (2003) and Gordy (2000) and a decreasing relationship in Lopez (2004) and, of course, BCBS (2006a). For the relationship with Size, these results, in their general rejection of decreasing asset correlations with decreasing Size, run counter to the relationship programmed into the Basel II Corporate exposure class IRB function for SME borrowers, while the lack of a programeed relationship for the Retail-Other asset class suggests the recognition of a potential absence of such a relationship for the smallest borrowers. Following Duellmann and
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Scheule (2003), we frame our results on Size patterns with several hypotheses on the relationship found in the literature.
In addition to these patterns, asset correlations estimated in this Chapter will also be characterized by generally low values. However, these values correspond in scale to those found in the literature. Frye (2008) and Chernih, Henrard, and Vanduffel (2010) demonstrate that asset correlations derived from loss data – such as that employed here – consistently generate values to scale with those found in this Chapter, while asset correlations derived from market equity data generally produces asset correlation values on scale with those applied within Basel II.
Applying these correlations within a single factor portfolio credit risk framework, we explore capital charge patterns along risk and size segmentations. Our results will show that the overall value of measured asset correlations can have a significant impact not only on the overall portfolio capital charge but also on capital charge patterns by segment.
In order to test and adjust for the results obtained with internally calibrated asset correlations, we apply a log odds adjustment to estimated asset correlations rendering their overall portfolio value on par with that obtained in Basel II. Our results show that for asset correlations at the scale of those found in Basel II, SME capital charges by Size should display a decreasing pattern. As a final note, however, we echo Duellman and Scheule (2003, p. 21) and observe that macro-prudential factors, as well as micro-
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prudential ones, play an important role in the preset calibration of Basel II parameters and capital charges. In this Thesis, these factors, e.g., the avoidance of pro-cyclical effects and the encouragement of looser credit conditions for small borrowers, are not considered as we focus on the micro-prudential factors reflected in the credit characteristics of an SME portfolio.
More specifically, Chapter 4 is organized as follows: In Section 4.1 we expand on the asymptotic single factor origins of the IRB risk weighting function, examined in Chapter 3 and Appendices A and B, and introduce a single factor model used for the estimation of SME correlations in our portfolio. In addition, a Monte Carlo simulation procedure is built around the model allowing for the non-asymptotic estimation of portfolio credit risk for our portfolio. In Section 4.2 we build on the work done in Chapters 2 and 3 by adding alternative dual-dimension segmentations. These segmentations are used in the estimation of PD, PD volatility and both asset and default correlations. Correlations here are estimated using the model introduced in Section 4.1. In Section 4.3, PD and correlation estimates derived in Section 4.2 are applied in both the asymptotic and non- asymptotic single factor models. The results, presented in this way, provide an interesting avenue of study when compared to each other and the results obtained in Chapter 3. Finally, Section 4.4 presents the Chapter’s conclusions.
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Section 4.1. A Single Factor Model for Portfolio Credit Risk
Recalling Section 3.1, Equation (3.1) can be reformulated as:
[ ( ) ] [ ( ) ( ) ( ) ] ( ) where, ( ) [ ( ) √ ( ) √ ] ( )
and, . Equation (4.2) is commonly referred to as the Vasicek distribution or function, as given in Equation (B.7). In Appendix B, a full derivation of Equation (4.2) is given. As previously noted, the Basel II IRB model is based on the Vasicek (2002) asymptotic approximation of the single risk factor model based on Merton (1974). In this Chapter we use the underlying Merton (1974) framework in two capacities: as a model for the estimation of portfolio credit risk loss distributions; and as a tool for the measurement and estimation of default correlations within a given credit portfolio.
In the first capacity, and in contrast to Chapter 3, we present the IRB risk weighting function as an asymptotic version of a single factor asset value model (AVM) for the determination of portfolio credit risk. In this asymptotic framework idiosyncratic risks are assumed away. We also present the framework as a non-asymptotic single factor
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asset value model for portfolio credit risk in which a Monte Carlo simulation procedure is used to generate portfolio loss distribution, and in which systematic and idiosyncratic risks in the portfolio are explicitly modelled; see Subsection 4.1.1.
In the second capacity, we use the framework, as presented in Gordy (2000), along with internally estimated PDs and PD volatilities, to non-parametrically estimate internally- calibrated asset (and, by extension, default) correlations for our SME portfolio; see Subsections 4.1.2 and 4.2.2. These correlations and their relationship to other credit risk measures and factors (e.g., PD, Size, etc…) are then evaluated and compared to values and relationships programmed into the Basel II IRB framework; see Section 3.1. We compare results generated using the internally calibrated correlations and both the asymptotic and non-asymptotic versions of this framework; see Section 4.3.
More specifically, asset correlation values, estimated by borrower segment, denote the dependence of borrowers in a given segment to a single underlying latent factor. Borrowers in different segments are allowed to have divergent dependencies on the same single factor, while borrowers in the same segment share the same dependence. Segments are defined along Risk and Size dimensions. Proceeding in this manner, we are able to test Basel II assumptions on the relationship between asset correlation values and Size, and asset correlation values and riskiness (as represented by the segment PDs).
Having generated our internally calibrated asset correlations, we define a non-asymptotic single factor model for portfolio credit risk. The internally calibrated asset correlations
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are used within this framework and a Monte Carlo simulation procedure is used to generate a portfolio loss distribution for our portfolio of SME borrowers.
In Subsection 4.3.2, we input the internally estimated asset correlations from Subsections 4.2.2 and 4.2.3 into an asymptotic single factor model. The capital charges generated within this asymptotic framework are contrasted against those obtained in the simulation- based non-asymptotic framework. This comparison technique provides us with an estimate of the capital estimation error, in an SME setting, due to the application of the asymptotic framework to a real-world portfolio of finite granularity. This work is similar in technique to that used in Tarashev and Zhu (2007).
Finally, in addition to a comparison of overall capital charges for the portfolio, we examine two simple and commonly applied capital allocation schemes within our non- asymptotic framework and compare them to the allocations under the asymptotic framework. In order to ensure efficient comparability, allocation schemes are applied to the same simulation-based VaR value.
Our results will indicate that the portfolio characteristics of a real-world portfolio with SME characteristics – high PD, low correlation values – may display a higher granularity effect when asset correlations are estimated from default data. In addition, we will find that single exposure size is an important factor in determining capital charges, one that is not properly accounted for in some analytical and asymptotic allocation schemes; see, for example, Heitfield, Burton, and Chomsisengphet (2006). In particular, we will challenge
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the inclusion of size effects through correlation parameters, as is done in the Basel II IRB framework.